ARTICLESPUBLISHED ONLINE: 15 AUGUST 2016 | DOI: 10.1038/NPHYS3842

Anti-parity–time symmetry with flying atomsPeng Peng1, Wanxia Cao1, Ce Shen1, Weizhi Qu1, JianmingWen2*, Liang Jiang2* and Yanhong Xiao1,3*The recently developed notion of parity–time (PT) symmetry in optical systems has spawned intriguing prospects. So far,most experimental implementations have been reported in solid-state systems. Here, we report the first experimentaldemonstration of optical anti-PT symmetry—the counterpart of conventional PT symmetry—in a warm atomic-vapour cell.Rapid coherence transport via flying atoms leads to a dissipative coupling between two long-lived atomic spin waves, allowingfor the observation of the essential features of anti-PT symmetry with unprecedented precision on the phase-transitionthreshold, as well as refractionless light propagation. Moreover, we show that a linear or nonlinear interaction between thetwo spatially separated beams can be achieved. Our results advance non-Hermitian physics by bridging to the field of atomic,molecular and optical physics, where new phenomena and applications in quantum and nonlinear optics aided by (anti-)PTsymmetry could be anticipated.

Canonical quantum mechanics postulates HermitianHamiltonians, to describe closed physical systems, to ensurereal eigenvalues and the orthonormality between eigenstates

with different eigenenergies. For open systems, non-HermitianHamiltonians with complex eigenvalues and non-orthogonaleigenfunctions are commonly expected. However, such anunderstanding has been radically challenged since a counterintuitivediscovery by Bender and Boettcher in 19981, where a wide class ofnon-Hermitian Hamiltonians (H ), subject to [H , PT ]=0 with theparity–time (PT) operator (PT ), could display entirely real spectrabelow some threshold. More strikingly, a sharp, symmetry-breakingtransition occurs once a non-Hermitian parameter crosses anexceptional point. This pioneering work immediately stimulatedconsiderable theoretical efforts in extending Hermitian quantumtheory to non-Hermitian systems2–4. Unfortunately, quantummechanics is, by nature, Hermitian and, thus, any attempt toobserve PT symmetry under such a theoretical framework is outof reach. Thanks to the mathematical isomorphism between thequantum Schrödinger and paraxial wave equations, a PT-symmetriccomplex potential can be readily established by judiciously makinguse of refractive indices with balanced gain and loss5 in an opticalsetting. This suggests optics as a fertile ground for experimentalinvestigations on PT symmetry without introducing any conflictwith standard quantum mechanics. Optical realizations6–14 havemotivated various synthetic designs with peculiar propertiesotherwise unattainable in traditional Hermitian structures,including band merging5, double refraction15, unidirectionalpropagation7,10,16–18, and power oscillation7,19. Other designs includecoherent perfect laser absorbers11,20,21, optical switches22, opticalcouplers23, and single-mode amplifiers14,24. Inspired by opticalsettings, systems using plasmonics25, LRC circuits26, acoustics27,artificial lattices28 and optomechanics29 have also been reported.

As a counterpart, an anti-PT-symmetric Hamiltonian followsH , PT = 0. Mathematically, a conventional PT-symmetricHamiltonian would become anti-PT-symmetric on multiplyingby ‘i’, implying properties of anti-PT systems conjugate to thoseof PT systems. For instance, in the symmetry-unbroken regime,lossless propagation in a PT system corresponds to refractionless(or unit-refraction) propagation in an anti-PT system. Such anintriguing effect may open up new opportunities for manipulating

light and form a complementary probe in non-Hermitian optics.Despite the appealing features, the realization of anti-PT symmetryis both theoretically and experimentally challenging. We noticethat the first theoretical proposal30 relies on a composite system ofmetamaterials by demanding an impractical balance of positive andnegative real refractive indices in space. A subsequent extensionconsiders an optical lattice of spatially driven cold atoms31. Notably,theoretical proposals32–34 using cold atoms for PT symmetry havealready been put forward in the spatial domain. Indeed, coherentlyprepared multilevel atoms are attractive systems for exploring non-Hermitian physics, because of their easy reconfiguration, flexibletunability, and especially the various coherence control techniquesenabled by electromagnetically induced transparency (EIT)35–40.However, such an experiment has yet to be performed, primarilydue to the hurdle of creating PT- or anti-PT-symmetric opticalpotentials through the rather complicated spatial modulation ofdriving fields or optical lattices.

In this article, we report the first experimental realization of anti-PT-symmetric optics by introducing a novel coupling mechanism.The coupling between two spatially separated probe fields ismediated through coherent mixing of spin waves created in twoparallel optical channels, in contrast to existing PT experimentswhere two optical fields of interest are coupled directly. Two spinwaves here are mixed through coherently diffusing atomic ground-state coherence carried by moving atoms. The spin-wave couplingenables either linear or nonlinear interactions between the twoprobe fields. Also, the precise measurement on (anti-)PT phasetransition in the frequency domain differentiates our work fromformer work rooted on either solid-material or (theoretical) cold-atomic systems. The characteristic of easy controllability furtherrenders the scheme as an alternative stage for probing non-Hermitian physics related to the exceptional point41–46.

Our experiments (schematically shown in Fig. 1a) are carried outin a 87Rb vacuum vapour cell of cylindrical shape, with a diameterof 2.5 cm and length 5 cm. The inner surface of the cell is coatedby coherence-preserving paraffin47, which allows atoms to undergothousands of wall collisions with little demolition of their internalquantum state. For comparisons between experiment and theory, anoptically thin medium is preferable with the cell temperature set at∼40 C. The cell is housed within a four-layer magnetic shield to

1Department of Physics, State Key Laboratory of Surface Physics, and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education),Fudan University, Shanghai 200433, China. 2Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA. 3CollaborativeInnovation Center of Advanced Microstructures, Nanjing 210093, China. *e-mail: jianming.wen@yale.edu; liang.jiang@yale.edu; yxiao@fudan.edu.cn

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS3842

2 2

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Figure 1 | Anti-PT-symmetric optics via rapid atomic-coherence transportin a warm 87Rb vapour cell. a, Schematic of a three-dimensional view of thesystem. Two spatially separated optical channels (Ch1 and Ch2) eachcontain collinearly propagating weak-probe and strong-control fieldsoperating under the condition of EIT. Ballistic atomic motion in aparan-wall-coated vacuum cell distributes atomic coherence through theoptically thin medium and results in eective coupling between the twooptical channels. With the temperature maintained at 40 C, the cell ishoused within a four-layer magnetic shield to screen out the ambientmagnetic field. Output probe transmission spectra are measured bysweeping a homogeneous magnetic field generated by a solenoid inside theshield. PD, photodetector. b, The implementation of anti-PT symmetryutilizing standard three-level3-type EIT configurations in two channels,where the two ground states are Zeeman sublevels |F=2,mF=0〉 and|F=2,mF=2〉, and the excited state is |F′= 1,mF′ = 1〉 of the 87Rb D1 line. Anexternal cavity diode laser provides the light for the probe and control fieldswith orthogonal circular polarizations. Right-circularly polarizedstrong-control fields,Ω(1)

1 andΩ(2)1 in Ch1 and Ch2 are, respectively, on

resonance with the transition |1〉→|3〉; while left-circularly polarizedweak-probe fields,Ω(1)

2 andΩ(2)2 in Ch1 and Ch2, are near resonant with

|2〉→|3〉, but frequency shifted in opposite directions by the same amount|∆0|, with∆0 being the dierence between the probe and the control fieldfrequencies. At steady state, the population is mainly in the groundstate |2〉.

screen out the ambient magnetic field. Inside the shield a solenoidis used to generate a uniform magnetic field, inducing a Zeemanshift δB to the two-photondetuning. The laser beam, derived fromanexternal cavity diode laser operating at the 87Rb D1 line (795 nm), isspatially split into four beams (1.2mm in diameter) using half-waveplates and polarization beam splitters (PBS). Orthogonally linearlypolarized probe and control beams are recombined, converted tocircular polarization by quarter-wave plates (QWP), and directedinto two optical channels, Ch1 and Ch2, separated transversely by1 cm. In the first experiment (Fig. 1b), two right-circularly polarizedstrong-control fields are resonant with the transition |1〉→|3〉, andtwo left-circularly polarized weak-probe fields are nearly resonantwith |2〉→|3〉, but frequency shifted in opposite directions by thesame one-photon detuning |∆0| using acousto-optical modulators(AOMs). To stabilize the phase relationship between the probe andcontrol beams, all AOMs are driven by oscillators phase locked toeach other. In each channel, the co-propagating probe and controlfields set up the standard 3-type EIT effect and create a long-lived ground-state coherence, with a lifetime of ∼100ms. The twospin waves are naturally coupled through the ballistic motion of

87Rb atoms in the cell, where the randomness and irreversibilityin the motion lays the foundation of the non-Hermiticity of theeffective Hamiltonian.

The atom–light interaction in our driven system is described bythe density-matrix formalism48 (equations (4)–(17) in Methods).Under certain approximations, the following time-dependent non-Hermitian (Floquet) Hamiltonian Heff is obtained to govern thedynamics of the two collective spin-wave excitations:

Heff=−δBI+H , where H=[|∆0|− iγ ′12 iΓce−2i|∆0 |t

iΓce2i|10 |t −|∆0|− iγ ′12

](1)

Here, I is a 2× 2 identity (or unit) matrix, γ ′12 = γ12 + Γc + 2Γp,with γ12 the dephasing rate of the ground-state coherence, Γc theground-state-coherence coupling rate between the two channels,and 2Γp=2|Ω1|

2/γ31 the total pumping rate by the two controlbeams with the same Rabi frequency Ω1, where γ31 is the atomicoptical coherence decay rate. We note that the determinant ofHeff, Det[Heff], is the denominator of the two coupled ground-state coherences−[γ ′12+ i(|∆0|−δB)][γ

′

12+ i(−|∆0|−δB)]+Γ2c (see

equations (18) and (19) in Methods). The resonance condition ofthe coupled system is met when Det[Heff] = 0, whose solutions,coinciding with the two eigenvalues of H , define the appearance ofthe two eigen-EIT supermodes (in terms of δB), with their real andimaginary parts being the line centres and linewidths, respectively.Experimentally, δB is a parameter that is swept to extract theeigenvalues of H . From another point of view, δB can be consideredas a common offset to the diagonal elements of H ; in turn, Hcan be mathematically produced from Heff by choosing a differentfrequency reference, indicating that, without the −δBI term, Hrecovers the essential physics. As such, H will be our starting pointthroughout this work.

For simplicity, we consider the Hamiltonian H in equation (1)at periodically distributed discrete time points satisfying e2i|∆0 |t=1.This reduces H to a simpler form:

H ′=[|∆0|− iγ ′12 iΓc

iΓc −|∆0|− iγ ′12

](2)

with its eigenvalues corresponding to the two eigen-EIT supermodes

ω±=−iγ ′12±√∆2

0−Γ2c (3)

Here, the real parts of ω± are the values of δB which correspond tothe resonance centres of the two coupled-EIT eigenmodes, and theimaginary parts represent their linewidths. As the essential results ofthis work, equations (1)–(3) contain rich physics and possess manyinteresting properties. First of all, for this 2×2 matrix, H satisfiesPTH=−H (that is, H , PT =0) in contrast to PTH=H (that is,[H , PT ]=0) for conventional PT symmetry; hence, we termH anti-PT-symmetric. In addition, this Hamiltonian leads to an intriguingphase transition exhibited on the two eigen-EIT spectral branches.Specifically, in the symmetry-unbroken regime (|∆0|<Γc), thetwo eigen-EIT resonances coincide at the centre δB=0, but withdifferent linewidths. The anti-PT symmetry breaking occurs at theexceptional point |∆0| = Γc where the two supermodes perfectlyoverlap. When |∆0|>Γc, the driven system enters the symmetry-broken regime, and the resonances bifurcate and exhibit level anti-crossing, resembling a passively coupled system.

The spectral profiles of the two eigen-EIT spectra can beexperimentally probed by slowly sweeping the magnetic field withtime and measuring the weak-probe transmission. Therefore, themeasured probe transmission is a function of both δB (lower x-axis)and time (upper x-axis), as shown in Fig. 2a1,a2 and b1,b2. Asexpected from equation (1), the EIT spectra display beating patterns

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Figure 2 | Representative transmission spectra of output probe light in anti-PT symmetry. a1,a2, Typical transmission spectra of output probe light afterCh1 (with the probe red-detuned from the control by∼15 Hz) and Ch2 (with the probe blue-detuned from the control by∼15 Hz) in the regime of anti-PTunbroken phase, exhibiting a beating frequency of∼30 Hz. b1,b2, Typical transmission spectra of output probe light with probe detuning∓60 Hz in Ch1and Ch2, respectively, in the regime of broken anti-PT phase. The bold, dotted curves in a1,a2,b1,b2 are the EIT spectra extracted at time points t whichsatisfy ei(2|∆0|t+1ϕ)= 1, and the beating notes marked in dark blue are for the phase reference (Methods). The insets are calculated curves from themodified theory (Methods). a3,b3, Comparison of the dotted curves in Ch1 (a1 and b1) and Ch2 (a2 and b2) plotted together. a4,b4, Uncoupled EITspectra separately measured from Ch1 (with red detuning∼15 Hz/∼60 Hz) with both channels’ control and only Ch1’s probe input on, and from Ch2 (withblue detuning∼15 Hz/∼60 Hz) with both channels’ control and only Ch2’s probe input on. The experimental parameters here are control powers of∼180 µW and probe powers of∼3.7 µW.

(light-blue curves in Fig. 2a1,a2 and b1,b2) oscillating at a frequency|2∆0|. The physical origin of the beating can be understood asfollows. The simultaneous presence of the oppositely detunedprobes in Ch1 and Ch2 locally creates two spin waves withinthe laser-beam volumes that precess, respectively, with e−i|10 |t andei|10 |t in a common rotating frame. Meanwhile, the atomic thermalmotion, at a much faster timescale than that needed for steady-statecoherence formation, redistributes and mixes the two spin wavescoherently within the entire vapour cell. The total coherence in each

channel is thus the sum of the two spin waves evolving at ei|∆0 |t ande−i|∆0 |t . Consequently, its amplitude ismodulated at frequency |2∆0|,giving rise to beating in the probe transmission spectra. Figure 2shows two sets of representative probe transmission spectra with∆0=∓15Hz (Fig. 2a1,a2) in the symmetry-unbroken regime and∆0=∓60Hz (Fig. 2b1,b2) in the symmetry-broken regime. Fromthe beating spectra, one can extract a set of EIT spectra, as denotedby the red/blue dots, for a particular value of ei(2|∆0 |t+1ϕ), where 1ϕis the difference between the two channels’ input probe–control

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS3842

2| 0| (Hz)

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Figure 3 | Anti-PT supermodes in coupled-EIT channels in a homogeneous, warm 87Rb vapour cell. a,b, Characteristics of the real part Re[ω] (a) andimaginary part Im[ω] (b) of the two eigenfrequencies of the coupled-EIT supermodes as a function of the probe detuning |∆0|. The data points areobtained from curve fitting the measured transmission spectra to the theoretical result. In a, as a comparison, the blue squares represent the EIT peakseparation between two uncoupled channels, and the red dots are for the coupled case with both Ch1 and Ch2 on. In b, the green dots and blue squares arethe extracted linewidths of the two eigen-EIT modes, respectively. The error bars are standard deviations obtained from ten measurements. As described inthe text, the linewidth values here are in good agreement with an independent check. The experimental parameters here are the same as those in Fig. 2.

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Figure 4 | Theoretical calculation and experimental demonstration of refractionless propagation of light (n=1+Re[χ]/2=1) assisted by anti-PTsymmetry in the symmetry-unbroken regime. a,b, Theoretical results. In the presence of two magnetic fields (giving a change of∆B to the two-photondetuning) with opposite signs in Ch1 and Ch2, in the uncoupled case, with only the linearly polarized beam in one channel on, the left-circularly polarizedcomponent in Ch1 (blue) and that in Ch2 (red) experience nonzero Re[χ] at δB=0 (a). In contrast, when both channels are on, they see zero Re[χ]at δB=0 when the coupled system is operated in the symmetry-unbroken region (b). c,d, Experimental results. In the presence of two artificially createdmagnetic fields with opposite signs in Ch1 and Ch2, the linearly polarized light in Ch1 and Ch2 experiences nonzero and reversed Faraday rotations(proportional to Re[χ]) at δB=0, when only one channel’s linearly polarized beam is on (uncoupled case) (c). When both the linearly polarized beams areon (coupled case), the rotation angle becomes zero at δB=0 for both channels, in good agreement with the theoretical prediction (d). The laser power foreach linearly polarized light beam is 140 µW. The two circularly polarized o-resonant beams with opposite helicity in the two channels to create thefictitious magnetic field have powers of 150 µW, and are red-detuned from the |F=2〉→|F′= 1〉 transition by 1.3 GHz. The beam size for all lasers isapproximately 1.2 mm in diameter and∆B= 15 Hz.

relative phases (see Methods). The spectra are fitted (red/blue lines)by a weighted sum of the two eigen-EIT spectra (see equations (18)and (19) in Methods) to give the linewidth and line centre values aspredicted by equation (3). As one can see, both the experimentalEIT spectra with the beat and the extracted spectra show goodagreement with the corresponding theoretical simulations (insets).

For ∆0 =∓15 Hz, the two eigen-EIT spectra (Fig. 2a3) coalesceat δB = 0, while for ∆0 =∓60 Hz, the two spectra (Fig. 2b3) arepulled closer to each other without complete overlap, compared tothe uncoupled EIT spectra (Fig. 2a4 and b4) separately measuredfrom Ch1/Ch2 with both control fields on but only Ch1’s/Ch2’sprobe on.

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1.0

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Figure 5 | Observation of interference between two EIT channels by manipulating the relative-phase dierence among laser beams. For |∆0|=0, whenall the beams in both channels are on, the EIT spectra of the output probes are measured for dierent relative phases of the input optical fields,1ϕ=1ϕ1−1ϕ2, where1ϕi is the probe–control relative phase in the ith channel. Depending on1ϕ, the interference eect may change the EIT spectradramatically. a1–f1, Experimental results, a2–f2, results calculated from the modified theory (see Methods), and a3–f3, Monte Carlo simulations (seeMethods), for1ϕ=0,π/2,3π/4,π,5π/4,3π/2. In a2–f2 and a3–f3 the plots are of the theoretical probe gain coecients (negative value meansabsorption), proportional to the imaginary part of the probe’s susceptibility. Experimental parameters: control powers of∼155 µW and probe powers of∼3.7 µW. Parameters used in the theoretical calculations: γ12=2π×7.5 Hz, Γc=2π× 15 Hz, γ13=2π×500 MHz, |Ω(1,2)

1 |=2.4× 10−4γ13, |Ω(1,2)2 |=

5× 10−5γ13. Parameters used in the Monte Carlo simulations: |Ω(1,2)1 |=2π× 1.0 MHz, |Ω(1,2)

2 |=0.1×|Ω(1,2)1 |, γ13=2π×500 MHz, the attenuation factor of

the ground-state coherence and population dierence upon each wall collision is e−(1/250) (equivalent to γ12=2π×7.5 Hz), the laser-beam size and cellsize are set to be the same as the experimental values.

The evolution of the anti-PT supermodes is carefully examinedby varying |∆0|. By applying the method elaborated in Methods,the extracted line centres and their corresponding linewidthsare plotted as a function of |2∆0| in Fig. 3, showing excellentagreement with theoretical predictions. Remarkably, the currentsystem has high resolution of the phase-transition threshold atthe Hz level. The experimental data (Fig. 3) evidently reveals theexceptional point occurring at |2∆0| = 30.5 Hz, which implies Γcof about 15Hz. According to equation (3), when ∆0 = 0 the fulllinewidth difference between the two supermodes should be 4Γc.This is verified from the fitting curve to the experimental datain Fig. 3b. Besides, equation (3) predicts the full linewidth ofthe lower eigenmode near ∆0= 0 to be 2(2Γp+ γ12). To confirmthis prediction experimentally, we first allow only one channel

on and obtain γ12 = 5 Hz through the zero-power EIT linewidthmeasurement. By utilizing the full width of 2(Γp+γ12)=69Hzof the single-channel EIT spectrum (not shown), we deduce2Γ p=59Hz and 2(2Γp + γ12) = 128 Hz, which agrees with thewidth of the lower eigenmode for ∆0 = 0 shown in Fig. 3b. Wenote that Γc is mainly determined by experimental parameterssuch as the laser power and beam size, cell geometry, and atomicthermal velocity.

As illustrated above, in the regime of unbroken anti-PTsymmetry, the initially separated EIT resonance centres of thetwo channels coalesce due to dissipative coupling. According tothe Kramers–Kronig relation, this coalescence of spectra shouldresult in zero Re[χ ] (real part of the total linear susceptibility)at the overlapped EIT centre—that is, unit refractive index n

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS3842

according to the relation n= 1+ Re[χ ]/2 for an optically thinmedium. Such refractionless (or unit-refraction) propagation oflight is an exact analogue of the lossless propagation of light in aPT-symmetric system in its symmetry-unbroken regime, whereprobe powers coherently oscillate between two coupled subsystemswithout experiencing either gain or loss. To demonstrate the effect,we inject linearly x-polarized laser beams into both Ch1 andCh2, resonant with the 5S1/2|F=2〉→5P1/2|F ′=1〉 transition, forwhich two circular polarization components form EIT. Instead ofintroducing ±|∆0| to the two-photon detuning in two channels,we here produce opposite Zeeman shifts ±∆B by effectivelycreating a local fictitious magnetic field through the a.c. Starkshifts of additional far-off-resonant circularly polarized lasers withopposite helicities in Ch1 and Ch2. At the output, for differentδB we measure the transmitted intensity difference between the45 and −45 polarization components (corresponding to the Jycomponent on the Poincaré sphere for light), which is the Faradayrotation signal and proportional to Re[χ ] for the left-circularlypolarized component. When only one linearly polarized beam ison (uncoupled case), the curves of the Faraday rotation versus δBare slightly shifted in opposite directions, resulting in nonzero andopposite rotations at δB= 0 in the two channels (Fig. 4c). In thecoupled case with the linearly polarized beams in both channels onand ∆B<Γc, the atomic-coherence coupling now gives rise to zeroFaraday rotation at δB= 0 (Fig. 4d). The experimental result is ingood agreement with the theoretical prediction (Fig. 4a,b).

Phase sensitivity arising from coherence transport is anotherintriguing feature of our system. We illustrate this for ∆0 = 0(Fig. 1) when the ground-state coherences in the two channels sharethe same oscillation frequency. By tuning 1ϕ=1ϕ1−1ϕ2, with1ϕ1 (1ϕ2) being the relative phase between the input control andprobe in Ch1 (Ch2), nontrivial interference phenomena can occurbetween the two channels. Let us recall that, at steady state, theconventional 3-type EIT system36 is phase insensitive, since theprobe transmission is independent of the relative phase betweenthe input control and probe. Due to the coupling of the two spinwaves, in contrast, the coupled-EIT system here becomes phasesensitive. As an example, Fig. 5 shows the output probe’s EIT spectraof the two channels for various 1ϕ. Specifically, as 1ϕ = π, theEIT amplitudes (Fig. 5d1) are reduced to zero because of completedestructive interference between the two channels, in contrast to themaximal EIT amplitude for 1ϕ= 0 (Fig. 5a1). In the interveningregion the dispersive feature appears (Fig. 5b1–c1 and e1–f1),generally associated with many phase-sensitive processes. Theexperimental data is confirmed both by our simplified analyticaltheory (Fig. 5a2–f2) and the Monte Carlo simulations (Fig. 5a3–f3,Methods), suggesting an agreement between the two theoreticalapproaches. The observation of such nonlocal interference impliesthat two spatially separated light beams can interfere with eachother, contrary to conventional interference experiments, where thelight beams must overlap.

The anti-PT system proposed here further provides a versatileway for photon–photon interactions. Despite the configuration ofcontrol/probe fields in Fig. 1b permitting linear interaction betweenthe two weak-probe fields, this interaction can be modified tobecome nonlinear by choosing the pumping configuration depictedin Fig. 6a. Here, the effective Hamiltonian governing the ground-state-coherence coupling between two channels takes a similar formto equation (1) and remains anti-PT-symmetric, if the two-photondetunings in the two channels are properly set. Interestingly, theresulting nonlinear process is analogous to the well-known four-wave mixing (FWM)49, and the two probes play similar roles tothose of the Stokes and anti-Stokes fields in FWM. Intuitively, asthe atomic coherence is being created in Ch1, the population isdriven from |1〉 to |2〉; when this atom enters Ch2, the population isthen brought back to |1〉. The process is continuously boosted along

with the coherence transfer. As a result, both weak-probe fields arecoherently amplified subject to the phase-matching condition. Toconfirm the analysis, we have carried out Monte Carlo simulationsfor the simple case of zero probe detunings. In Fig. 6b, the probetransmissions from both channels are plotted as a function of δB,assuming the initial phases of all optical fields to be zero. We findgain in both channels around δB = 0. For large δB, although thephase-matching condition is still satisfied, the reduced ground-state coherence provides insufficient gain to compensate loss, whichindicates that coherence plays an essential role in this FWM-likeprocess. To further assess the nonlinear interaction, let us look at theprobe transmissions for δB=0 by varying the phases (θp1,θp2) of thetwo probe fields. The simulation is presented in Fig. 6c, where twoprobe phases are swept together (θp1= θp2) whilst the two controlphases (θc) are set at zero. Efficient FWM is expected to occurin our system when perfect phase matching is approached—thatis, θp1 + θp2 = 2θc + 2mπ with m an integer. Such an intuition isverified by the simulation shown in Fig. 6c, where the maximal gainindeed appears at θp1,p2=0, π, 2π. Complete theoretical modellingincluding nonzero probe detunings and the effects of the exceptionalpoint will be considered in the future. To the best of our knowledge,so far all demonstrations of EIT-assisted nonlinear optics are basedon optical nonlinearities inherent to suitable multilevel atomicstructures36. In comparison, here the effective nonlinearity is builtfrom coherence transport.

Given the tight connection of our system tomagnetometers, slowlight and quantummemory50, new directions may be opened up forprecisionmeasurements, quantum optics and quantum informationscience. The EIT spectra with beating is similar to Ramsey fringesand can thus be used for precisely measuring the optical phase orthemagnetic field. It is also intriguing to investigate the possibility ofusing the synthetic FWMconfiguration described above to entangletwo spatially non-overlapping weak lasers, and to explore the effectof the phase transition in anti-PT symmetry on quantum noise.Furthermore, with some modifications this system can be switchedto be PT-symmetric. Despite the mediator analysed in this workbeing freely moving atoms, similar strategies might be extended toother systemswhere the indirect coupling is established through, butnot limited to, electrons, polaritons, plasmas, or phonons.

MethodsMethods, including statements of data availability and anyassociated accession codes and references, are available in theonline version of this paper.

Received 25 September 2015; accepted 21 June 2016;published online 15 August 2016

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NATURE PHYSICS | VOL 12 | DECEMBER 2016 | www.nature.com/naturephysics

NATURE PHYSICS DOI: 10.1038/NPHYS3842 ARTICLES

0 π/2 π 3π/2 2πPhase of the two probes

−500 0 500

−1

0

1b

a

c

Prob

e ga

inco

effici

ent (

a.u.

)

−1

0

1

Prob

e ga

inco

effici

ent (

a.u.

)

B (Hz)δ

Control

Ch1 Ch2|3⟩ |3⟩

|1⟩ |1⟩|2⟩ |2⟩

1(1)Ω

Probe1(2)Ω

Probe2

(1)ΩControl

2(2)Ω

2Bδ

2Bδ

Figure 6 | Proposal for eective nonlinear interaction between twoweak-probe fields aided by anti-PT-symmetric coupling of spin waves.a, Light configuration used to realize a four-wave-mixing-like process.Dierent from Fig. 1b, the two control (probe) fields in the two channelshave opposite circular helicities, and are coupled to dierent transitions. Torealize the anti-PT-symmetric Hamiltonian (equation (1)) for the coupledspin waves, the two probe frequency detunings (not drawn) should havethe same sign, in contrast to that in Fig. 1b. The schematic in a is a specialcase with zero probe detuning∆0. b, Monte Carlo simulation of the twoprobe’s gain coecients, proportional to the imaginary part of the probe’ssusceptibility, as a function of δB. Here, positive (negative) values stand forgain (absorption). For simplicity, the initial phases of all laser fields are setto be zero. A maximal gain of 22% can be obtained in the currentexperimental conditions, and the linewidth of the gain curve here isdetermined by the overall decay rate γ ′12 of the ground-state coherence, asdescribed in equation (1). c, Monte Carlo simulation of two probetransmissions when the initial probe phases are synchronized and swepttogether, the initial phases of the two control fields are set at zero, andδB=0. The displayed phase sensitivity, originating from phase-matchingcondition, implies that the process is akin to a normal phase-sensitive FWMin nonlinear optics. Parameters used in the simulation: the Rabi frequenciesof the two controls (probes) take the same values as those in the MonteCarlo simulation in Fig. 5. All other parameters are also the same.

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AcknowledgementsWe are grateful to V. V. Albert for reading the manuscript. This work is supported byNational Key Research Program of China under Grant No. 2016YFA0302000, andNNSFC under Grant No. 11322436. J.W. and L.J. acknowledge funding support from theARO, the AFSORMURI, the ARL CDQI program, the Alfred P. Sloan Foundation, andthe David and Lucile Packard Foundation.

Author contributionsJ.W., L.J. and Y.X. conceived the idea. Y.X. supervised the project. P.P. performed theexperiment. W.C., P.P. and L.J. did the theoretical derivation and numerical calculationswith contributions from all other authors. J.W., L.J. and Y.X. wrote the manuscript withcontributions from all other authors. All contributed to the discussion of the project andanalysis of the experimental data.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints.Correspondence and requests for materials should be addressed to J.W., L.J. or Y.X.

Competing financial interestsThe authors declare no competing financial interests.

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1145

ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS3842

MethodsDerivation of the anti-PT-symmetric Hamiltonian. The system of interest isschematically depicted in Fig. 1a, where two optical channels 1 and 2 with nospatial overlap are each composed of a collinear weak-probe field and astrong-control field. The atomic level structure is the three-level3-type EITconfiguration shown in Fig. 1b. Two identical, right-circularly polarized controlfields with the same Rabi (Ω (1)

1 =Ω(2)1 ) and angular (ω(1)1 =ω

(2)1 ) frequencies are

applied on resonance with the atomic transition |1〉→|3〉; while two left-circularlypolarized probe fields (withΩ (1)

2 =Ω(2)2 and ω(1)2 6=ω

(2)2 ) are tuned almost resonant

with the transition |2〉→|3〉, but frequency shifted in opposite directions by a smallvalue |10|. Under the EIT conditions, the population is mainly distributed in |2〉. Acommon magnetic field is applied to implement the two-photon detuning in theEIT spectra measurement. The two-photon detunings in Ch1 and Ch2 areδ(1)=∆

(1)1 −∆

(1)2 =|∆0|−δB and δ(2)=∆(2)

1 −∆(2)2 =−|∆0|−δB, respectively, where

∆(j)1 (∆(j)

2 ) is the one-photon detuning of the control (probe) in the jth channel(j=1,2). The ground-state coherences in the two channels are effectively coupledthrough thermal motion at a rate Γc.

Under the rotating-wave approximation, the atom–light interaction can bedescribed by the following density-matrix formalism48,

ρ(1)11 =−iΩ

(1)1 ρ

(1)13 + iΩ

(1)∗1 ρ

(1)31 +γρ

(1)33 −γ

′(ρ(1)11 −ρ

(1)22

)(4)

ρ(1)22 =−iΩ

(1)2 ρ

(1)23 + iΩ

(1)∗2 ρ

(1)32 +γρ

(1)33 +γ

′(ρ(1)11 −ρ

(1)22

)(5)

ρ(1)31 =−

(γ13− i∆(1)

1

)ρ(1)31 + iΩ

(1)2 ρ

(1)21 + iΩ

(1)1

(ρ(1)11 −ρ

(1)33

)(6)

ρ(1)32 =−

(γ23− i∆(1)

2

)ρ(1)32 + iΩ

(1)1 ρ

(1)12 + iΩ

(1)2

(ρ(1)22 −ρ

(1)33

)(7)

ρ(2)11 =−iΩ

(2)1 ρ

(2)13 + iΩ

(2)∗1 ρ

(2)31 +γρ

(2)33 −γ

′(ρ(2)11 −ρ

(2)22

)(8)

ρ(2)22 =−iΩ

(2)2 ρ

(2)23 + iΩ

(2)∗2 ρ

(2)32 +γρ

(2)33 +γ

′(ρ(2)11 −ρ

(2)22

)(9)

ρ(2)31 =−

(γ13− i∆(2)

1

)ρ(2)31 + iΩ

(2)2 ρ

(2)21 + iΩ

(2)1

(ρ(2)11 −ρ

(2)33

)(10)

ρ(2)32 =−

(γ23− i∆(2)

2

)ρ(2)32 + iΩ

(2)1 ρ

(2)12 + iΩ

(2)2

(ρ(2)22 −ρ

(2)33

)(11)

ρ(1)12 =−

(γ12+ iδ(1)

)ρ(1)12 + iΩ

(1)∗1 ρ

(1)32 − iΩ

(1)2 ρ

(1)13

−(Γc+Γ

(2)p

)ρ(1)12 +Γce−i2|10 |tρ

(2)12 (12)

ρ(2)12 =−

(γ12+ iδ(2)

)ρ(2)12 + iΩ

(2)∗1 ρ

(2)32 − iΩ

(2)2 ρ

(2)13

−(Γc+Γ

(1)p

)ρ(2)12 +Γcei2|10 |tρ

(1)12 (13)

ρ(1)11 (0)+ρ

(1)22 (0)+ρ

(1)33 (0)=1 (14)

ρ(2)11 (0)+ρ

(2)22 (0)+ρ

(2)33 (0)=1 (15)

Here, γij represents the decay rate of the coherence between states |i〉 and |j〉, γ ′stands for the decay rate of the ground-state population difference, γ is the decayrate of the excited state, and Γ (j)

p =(|Ω(j)1 |

2/γ23) denotes the optical pumping ratefor EIT in the jth channel.

Since the optical coherences ρ(j)31 and ρ(j)32 decay much faster than the

ground-state coherences ρ(j)12 , we can assume that they follow the slow oscillationsin the ground-state coherences. Therefore, by setting the time derivatives of opticalcoherences in equations (6) and (7) and (10) and (11) to be zero, one can expressthe optical coherences in terms of ground-state coherences. The control beams inthe experiment are relatively weak such that the excited-state population ρ(j)33 =0.Also, we assume thatΩ (j)

1 Ω(j)2 , so that ρ(j)22 =1. By adopting a procedure similar to

that implemented in ref. 48, only equations (12) and (13) survive and play anessential role for describing the underlying physics. Thus, one can obtain thefollowing coupled equations of two collective spin-wave excitations (or in short,spin waves) associated with the ground-state coherences, ρ(1)12 and ρ(2)12 :

ρ(1)12 =−

(γ ′12+ iδ

(1))ρ(1)12 +Γce−2i|10 |tρ(2)12 −

Ω(1)∗1 Ω

(1)2

γ23(16)

ρ(2)12 =−

(γ ′12+ iδ

(2))ρ(2)12 +Γce2i|10 |tρ(1)12 −

Ω(2)∗1 Ω

(2)2

γ23(17)

Here, γ ′12=γ12+Γc+Γ(1)p +Γ

(2)p =γ12+Γc+2Γp represents the total effective

decay rate of the spin waves, with the assumption of equal control powers in Ch1and Ch2. Information about the collective spin-wave excitations can be simplydetected via EIT by measuring the weak-probe outputs after the cell, as their

transmittances are linear functions of the ground-state coherence in the opticallythin regime—that is, ρ32=(iΩ2+ iΩ1ρ12)/γ23.

From the coupled equations (16) and (17), the effective non-HermitianHamiltonian Heff (as shown in equation (1)) can be deduced that governs thedynamics of the two collective spin-wave excitations:

Heff=−δBI+H , where H=[|∆0|− iγ ′12 iΓce−2i|∆0 |t

iΓce2i|10 |t −|∆0|− iγ ′12

]It is straightforward to show H is anti-PT-symmetric because it satisfiesPTH=−H , in contrast to PTH=H in conventional PT symmetry. Anotherimportant feature of this equation is that H is symmetric under discrete timetranslations, t→t+π/ |∆0|, and is hence a periodic function in time,H (t)=H(t+π/ |∆0|) with the period π/ |∆0| of the perturbation. Such anobservation of the symmetry of the Hamiltonian (1) enables the use of the Floquetformalism, and could become another control in investigating the dynamics of thecoupled spin waves.

In the adiabatic limit, the atomic dynamics follows the effective Hamiltonian(1), which evolves slowly under the presence of the terms e±2i|∆0 |t . In thesteady-state approximation, the two collective spin-wave excitations take the form

ρ(1)12 =−

Ω(1)∗1 Ω

(1)2

γ23

(γ ′12+ iδ(2)

)−

Ω(2)∗1 Ω

(2)2

γ23Γce−2i|∆0 |t

(γ ′12+ iδ(1))(γ ′12+ iδ(2))−Γ 2c

(18)

ρ(2)12 =−

Ω(2)∗1 Ω

(2)2

γ23(γ ′12+ iδ(1))−

Ω(1)∗1 Ω

(1)2

γ23Γce2i|∆0 |t

(γ ′12+ iδ(1))(γ ′12+ iδ(2))−Γ 2c

(19)

The physics behind equations (18) and (19) is the following. Each spin-waveexcitation is a linear superposition of the two coupled-EIT eigenmodes, whosespectral profiles can be obtained by setting the denominator of equations (18) or(19) to zero. Each eigenmode has contributions from both channels and exhibits abeating frequency of 2 |∆0|.

Equations (16) and (17) can be also perturbatively solved in the non-adiabaticlimit. In such a case, in fact, one can expand the two spin-wave excitations in termsof Fourier series—that is, ρ(1)12 =

∑n=∞n=−∞ ρ

(1)12 (n)e2in|∆0 |t and

ρ(2)12 =

∑n=∞n=−∞ ρ

(2)12 (n)e2in|10 |t . Equations (12) and (13) can then be evaluated by

matching terms with the same order. For the range of parameters in the currentwork, even truncating the series up to the first order n=1 would yield a goodagreement between theory and experiment. After lengthy algebra, one can obtain:

ρ(1)12 =ρ

(1)12 (0)+ρ

(1)12 (−2 |∆0|)e−i2|10 |t

=1γ23

(Ω

(1)∗1 Ω

(1)2 (γ

′

12+ iδ(1))Γ 2

c −(γ′

12+ iδ(1))2 +

ΓcΩ(2)∗1 Ω

(2)2

Γ 2c −(γ

′

12+ iδ(2))2 e−i2|∆0 |t

)(20)

ρ(2)12 =ρ

(2)12 (0)+ρ

(2)12 (2 |∆0|)ei2|10 |t

=1γ23

(Ω

(2)∗1 Ω

(2)2 (γ

′

12+ iδ(2))Γ 2

c −(γ′

12+ iδ(2))2 +

ΓcΩ(1)∗1 Ω

(1)2

Γ 2c −(γ

′

12+ iδ(1))2 e

i2|∆0 |t

)(21)

We have proved that equations (20) and (21) can be converted back to equations(18) and (19) when the adiabatic condition is met. Our experiment was carried outin the adiabatic regime where∆0 and Γc are much smaller than γ ′12.

Criteria for phase reference selection. As can be seen from equations (18) and(19), to extract the linewidth and line centre of the eigen-EIT spectra from themeasured EIT spectra containing beat patterns, we need to identify all the pointson the beat with time t which give a fixed value to ei(2|∆0 |t+1ϕ), where1ϕ is thedifference between the two channels’ input probe–control relative phases. Forconvenience, we choose time points satisfying ei(2|∆0 |t+1ϕ)=1. To identify thesepoints, a reference point is needed such that all other points can be found at timeintervals that are integral multiples of the beating period from it.

Equations (20) shows that, for Ch1, the d.c. part of the spin wave has its centredetermined by the two-photon detuning of Ch1, while the centre of its a.c. part isdetermined by the two-photon detuning of Ch2. For the spin wave in Ch2 it is viceversa. Moreover, equation (20) implies that, for the Ch1 EIT spectra, only whenδB=−|∆0| (that is, δ(2)=0) do the atoms not impose an additional dynamicalphase on top of the beating term e−2i|∆0 |t . We note that the beat note at this locationhas the maximal amplitude across the whole spectral profile. Meanwhile, byexamining the corresponding optical coherence for Ch1’s probe field, we can seethat, at the peak of the maximal beating note, e−i(2|∆0 |t+1ϕ)=1 on the time axis issatisfied. Similarly, equation (21) indicates that, for Ch2, such a beat note is locatedat δB=|∆0|. Based upon this analysis, in the experiment we first identify the beatnote with the maximal amplitude and then choose its peak location as the timereference origin (see Fig. 2a1,a2 and b1,b2).

The phase criteria we used here are not the only possible choice, but aconvenient one. In fact, if the reference origin is offset from the peak of the

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NATURE PHYSICS DOI: 10.1038/NPHYS3842 ARTICLESmaximal beat note, the lineshape of the extracted coupled-EIT spectra would bemodified in some way. For our experimental conditions, for instance, in the regimewhere Γc> |∆0|, the degree of tilting in the lineshape of EIT will change, due to thechanged weight of the dispersive components; in the regime where Γc< |∆0|, twoequally weighted peaks in the extracted EIT spectra become unequal. Fortunately,such a situation does not affect the extracted values of the line centre and linewidthof the EIT eignemodes, but affects only the weights of the two eigenmodes and theratio between the Lorentzian and dispersive components. The unequal peaks(red/blue curves in Fig. 2b1,b2) and slightly tilted Lorentzians (red/blue curves inFig. 2a1,a2) indicate that the criteria stated above give a small deviation of thephase reference origin selected from the ideal value. This is mainly due to the finitenumber of beats available in the experiment as well as the small drift of input laserphases,1ϕ. For example, in the spectra from Ch1, the beat with the maximalamplitude is not precisely located at δB=−|∆0|. Such a small discrepancy isunavoidable and vanishes only for infinitely dense beat notes, which require aninfinitely long data acquisition time and absolutely no phase instability in any ofthe optical fields.

Modification of the model. Although the theory described above, named‘unmodified theory’ from now on, captures the essential physics behind theobserved anti-PT symmetry, a careful comparison of the experimentally measuredEIT spectra still reveals a discrepancy with respect to those computed from theunmodified theory. An example (shown in Supplementary Fig. 1) exhibits adisagreement between the unmodified theory (Supplementary Fig. 1 a2–d2) andthe experiment (Supplementary Fig. 1 a1–d1). In particular, the calculated beatingcontrast is much smaller than that measured experimentally. In addition, for thecase of∆0=0, EIT spectra computed (not shown) from equations (18) and (19) donot agree with the complete destructive interference effect observed for1ϕ=π inthe experiment (Fig. 5d1). This discrepancy occurs because we neglected coherencetransfer between the cell regions inside and outside the laser beams; in other words,the full role of motional averaging in the system was not taken into account. Inpractice, atoms fly into and out of the laser beams rapidly and, as a consequence,coherences produced from the two channels mix much better within the whole cellvolume than those described from the simple model presented above. Intuitively,this fast mixing should result in a higher beat contrast for∆0 6=0 and a bettercancellation of EIT at1ϕ=π for∆0=0.

Before addressing this issue, let us recall that the EIT lineshape in a buffer-gasor wall-coated vapour cell has a dual structure if the laser-beam size is muchsmaller than the cell volume38,51–54. The broad spectral structure comes from theone-time interaction of the atoms and the light, with a linewidth determined by thetransit time across the laser beam (∼300 kHz in our experiment). The narrowerstructure on top, due to the much longer time that the atoms spend outside the laserbeam experiencing dark evolution, has a linewidth (∼100Hz in our experiment)determined by the ground-state-coherence decay rate and the optical pumpingrate. Given the large difference in the two linewidths, the broad structure may beconsidered as a constant offset added to the coherence calculated by the simplemodel (equations (16) and (17)) in the frequency range of the narrow structure.

On the basis of the above analysis, we phenomenologically add an offset to theground-state coherence to include the process of coherence exchange between thevolumes inside and outside the laser beams. That is, we modify the source terms forthe coherences in equations (16) and (17) as follows:

ρ(1)12 =−

(γ ′12+ iδ

(1))ρ(1)12 +Γce−2i|∆0 |tρ(2)12 −

Ω(1)∗1 Ω

(1)2

γ23

(η+η

γ ′12+ iδ(1)

Γc

)(22)

ρ(2)12 =−

(γ ′12+ iδ

(2))ρ(2)12 +Γce2i|∆0 |tρ(1)12 −

Ω(2)∗1 Ω

(2)2

γ23

(η+η

γ ′12+ iδ(2)

Γc

)(23)

Here, the ((γ ′12+ iδ(1,2))/Γc) terms on the right-hand side give rise to an offset ofη((Ω

(1,2)∗1 Ω

(1,2)2 )/γ23Γc) to the coherences ρ(1,2)12 for the single-channel case (that is,

setting the coherence of the other channel ρ(2,1)12 to be zero). Alternatively, this termcan be understood as an additional source for ρ(1,2)12 contributed by the regionoutside the laser beams through coherence transport. The dimensionless constant ηis a fitting parameter. It is evident that this modification does not change theeffective Hamiltonian (1) of the system, because it affects only the source term forthe coherence. Interestingly, such a simple correction made in the model enables aquantitative comparison with experimental data.

Equations (22) and (23) can be solved using the perturbative approachdescribed above. The calculated EIT transmission spectra of the probe beams in thetwo channels are shown in Supplementary Fig. 1 a3–d3, which show an excellentagreement with the experimental measurements displayed in Supplementary Fig. 1a1–d1. When plotting the EIT spectra with no beat, the phase reference origin herehas been shifted by 1/30th of the beating period with respect to the above criteria,to match the shape of the EIT curve extracted from the experimental data (Fig. 2).In the calculations, we allow the magnetic field to vary slowly enough with time, asthe sweeping magnetic field does in the experiment. The time t and δB are depicted,respectively, on the upper and lower horizontal axes. We also verified that the

modified model shows an excellent agreement with the EIT interferenceexperiment (Fig. 5a2–f2).

Monte Carlo simulations. To further verify the correctness of our theoreticalmodel described above, we have carried out full Monte Carlo simulations.Previously, analytical models and two-dimensional Monte Carlo simulations havebeen developed to study repetitive atom–light interactions under EIT in bothcoated cells and buffer-gas cells, for a single-channel laser-beam geometry51–54. Wehere extend the Monte Carlo simulation to the current two-channel scenario. In thesimulation, for simplicity, a round area is assumed to represent the laser-beam crosssection. The atoms are allowed to move freely and collide with the cell wall, and theangular distribution relative to the surface normal of bounced atoms follows acosine distribution55. For each time that the atom bounces off the cell wall, toaccount for the ground-state population difference and coherence decay, we assignan attenuation factor to the population difference and the ground-state coherence,and then let the atomic dynamics continue until the system reaches its steady state.Typically, 5,000 trajectories are averaged to reduce the heavy computational load ofthe simulation. An example (Fig. 5a3–f3) is provided for the case where∆0=0. Aswe can see, the simulation matches the experiment very well.

Measurements and methods for curve fitting. As described above, thecoupled-EIT eigenspectra measured by the probe transmittance exhibit anoscillatory pattern at a beating frequency of 2 |∆0|. Experimentally, to properlyacquire such a beating spectrum, the following conditions should be satisfied. First,the magnetic field is swept slowly enough such that the atoms form steady-statebeat oscillations for each δB. Second, to prevent the beating from being washed outduring time averaging over several magnetic-field-sweeping periods, themagnetic-field-sweeping period is maintained at integer multiples of the beatingperiod. Third,1ϕ should not drift during each sweeping, otherwise the phasefluctuations may wash out the beating fringes during averaging. Therefore, aftereach scan, we recheck and make sure that1ϕ remains unchanged.

After acquiring the EIT spectra with the beat note, we locate the beat with themaximal amplitude (the bold dark-blue lines highlighted in Fig. 2a1,a2 and b1,b2),and set the position of this beat’s peak as the time reference origin. Starting fromthis reference origin, the remaining discrete time points (red/blue dots inFig. 2a1,a2 and b1,b2) are then sequentially identified at distances of integralmultiples of the beating period. As the phase of the beat depends on the relativephase between the two coherences (oscillating at e±i|∆0 |t respectively) as well as δB(see equations (20)–(21)), these identified time points (red/blue dots) are located atdifferent positions of the beat notes. The EIT spectra formed by these red/blue dotsare a weighted sum of the two eigen-EIT supermodes as given in equation (3),whose linewidth and line centre can be then obtained by curve fitting. Note that theEIT spectra of either channel is sufficient to extract the line centres and linewidthsof both eigenmodes, and each eigenmode is generally a tilted Lorentzian because itis composed of a Lorentzian and a dispersive component with the same linewidthand line centre, according to equations (18) and (19).

The two-channel EIT under coupling has a profile distinct from that withoutcoupling. In our definition, the uncoupled EIT spectra from Ch1 (Ch2) aremeasured with both channels’ control fields and only Ch1’s (Ch2’s) input probe on(see Fig. 2a4 and b4). In the experiment, we first turn on the control and probefields in only one channel and optimize the laser frequencies to ensure a symmetricEIT lineshape. The coupled EIT is measured when the probe and control in bothchannels are on. The separation between the two uncoupled EIT resonance centresis simply |2∆0|. When both channels are on, the extracted EIT spectral profilesdeviate from a single Lorentzian. Here, on the one hand, due to a small phase offsetas explained above, the existence of a small dispersive component slightly tilts theLorentzian in the regime where Γc> |∆0|. As a result, the extracted EIT spectrumin either channel is asymmetric with respect to δB. On the other hand, the lineshapein the regime where 0c< |∆0| has dual peaks because the centres of two EITeigenmodes do not overlap. In the regime where Γc> |∆0|, the extracted EITspectra have dual linewidths because they are a superposition of two EITeigenmodes with the same line centre but different linewidths. Indeed, we foundthat, without introducing a second (slightly tilted) Lorentzian with a differentlinewidth, a single-linewidth Lorentzian does not fit the experimental data well. Asdisplayed in Figs 2 and 3 in the main text, our theory is in excellent agreement withthe experimentally extracted data. The coupling rate Γc between the two channelsis deduced from the phase-transition point by tuning |∆0|, and the theoreticalmodel for calculating Γc is under development.

Data availability. The data that support the plots within this paper and otherfindings of this study are available from the corresponding authors uponreasonable request.

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Ramsey narrowing. Phys. Rev. Lett. 96, 043601 (2006).

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS3842

52. Xiao, Y., Novikova, I., Phillips, D. F. & Walsworth, R. L. Repeated interactionmodel for diffusion-induced Ramsey narrowing. Opt. Express 16,14128–14141 (2008).

53. Klein, M., Hohensee, M., Phillips, D. F. & Walsworth, R. L. Electromagneticallyinduced transparency in paraffin-coated vapor cells. Phys. Rev. A 83,013826 (2011).

54. Xu, Z., Qu, W., Gao, R., Hu, X. & Xiao, Y. Linewidth of electromagneticallyinduced transparency under motional averaging in a coated vapor cell.Chinese Phys. B 22, 033202 (2013).

55. Budker, D. et al . Microwave transitions and nonlinear magneto-opticalrotation in anti-relaxation-coated cells. Phys. Rev. A 71,012903 (2005).

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